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[IIVVP807860819 11 1111DEVELOPMENT OF A MATHEMATICAL MODEL FOR

BLOWDOWN OF VESSELS CONTAINING MULTI-COMPONENT HYDROCARBON MIXTURES'

A thesissubmitted to the University of London for the degreeof Doctorof Philosophy

by

SHAN MENG ANGELA WONG B. Eng (Hons)

Departmentof Chemical EngineeringUniversity College LondonTorrington PlaceLondon WC 1E7JEJune 1998

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ABSTRACTThis thesis describes the development of a mathematical model, BLOWSIM, forsimulating vapour spaceblowdown of an isolated vessel containing single (vapour)or two-phase (vapour and liquid) hydrocarbon mixtures based on three CubicEquations of State (CEOS). These include Soaves Redlich-Kwong (SRK), Peng-Robinson (PR) andthe recently developedTwu-Coon-Cunningham (TCC) CEOS.The performances of the above equations are first evaluated by comparing theirpredictions for a range of important thermophysical properties (includingvapour/liquid equilibrium data, speed of sound and fluid densities) with experimentaldata for single and multi-component hydrocarbon systems. These data are reportedas a function of reduced pressures and temperatures in the ranges 0.00053 - 43.41 and0.33 - 2.09 respectively. Typical systems tested include pure alkanes as well asmixtures containing methane, ethane, propane, H2S, CO2, N2 and trace amounts ofheavy hydrocarbons.The above is then followed by applications of all three equations in the blowdownmodel and comparing the results with those obtained from a number of experimentsrelating to the blowdown of the various hydrocarbon systems from a maximumpressure of 120 atm and ambient temperature. Typical output include the variationsof fluid pressure, temperature (both liquid and vapour), discharge rate as well as thewetted and unwetted wall temperatures with time.Another major part of the study includes investigating the effects of differentassumptions elating to the estimation of the liquid/wall heat transfer coefficient, thethermodynamic trajectory of the fluid in the vessel as well as the fluid phase at theorifice on blowdown predictions.We find that in general all three CEOS provide a similar level of accuracywith TCCCEOS providing the best performance in terms of predicting vapour speedof soundat Pr > 3. However, the equation gives rise to relatively large errors in predictingliquid speedof sound at Tr

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ACKNOWLEDGEMENTS

I would like to express my gratitude to the Crouch Foundation, the Society ofChemical Industry, the ORS Award scheme and the GraduateSchool of UniversityCollege London for their financial support for this work.My special thanks go to my supervisor, Dr Haroun Mahgerefteh for his excellentsupervision,constant encouragement,and unlimited patience during the course of myresearch.I would also like to thank Dr. Chorny H. Twu from Simulation Science,California,USA for many useful discussions relating to the work associatedwith equations ofstate.Additionally, I would like to acknowledge the help with computer programming fromHenry Tillotson of the Department of Information Systems at UCL.Finally, it has been a pleasure to receive the help of many members of theDepartment, n particular Martin Vale andhis colleaguesof the electronic workshop.

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LIST OF CONTENTS

ABSTRACT 1

ACKNOWLEDGMENTS 2

CHAPTER 1 INTRODUCTION 9

CHAPTER 2 VAPOUR SPACE BLOWDOWN PHENOMENA

2.1 INTRODUCTION

2.2 VAPOUR SPACE BLOWDOWN

1313

2.2.1 Heat Transfer between VesselWall and Vapour Space 17

2.2.2 Heat Transfer between VesselWall and Liquid Space 282.2.3 Heat and Mass Transfer Between Vapour and Liquid 33

Phases

2.2.4 Heat Transfer Between the Vessel and the Surrounding 34

2.2.5 Flow Through the Relief Valve 342.3 CONCLUSION

CHAPTER 3 REVIEW OF MATHEMATICAL MODELS FORBLOWDOWN SIMULATION

3.1 INTRODUCTION

36

383.2 TRADITIONAL ENGINEERING METHODS FOR BLOWDOWN 38

SIMULATION

3.3 SIMPLE MATHEMATICAL MODELS FOR BLOWDOWN OF 39NON-CONDENSABLE GASES

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3.3.1 Reynolds and Kays [195813.3.2 Byrnes et al. [196413.3.3 Montgomery [1995]

394448

3.4 RIGOROUSMATHEMATICAL MODELS FORBLOWDOWN 533.4.1 Blowdown [Haque et al., 1990; Haque et al., 1992a] 53

3.4.1.1 Haque et a!., 1990 53

3.4.1.2 Haque et al., 1992a 55

3.4.2 Split Fluid Model [Overa et al., 19941 663.5 CONCLUSION

CHAPTER4 THE DEVELOPMENT OF A BLOWDOWNMATHEMATICAL MODEL : BLOWSIM

4.1 INTRODUCTION

72

744.2 DEVELOPMENT OF BLOWSIM MATHEMATICAL MODEL 75

4.2.1 Application of Finite Difference Method to Blowdown 75Calculation

4.2.2 Fluid Phase Material Balances 76

4.2.2.1 Zone 1: Condensation: in Sub-Cooled Vapour 77

4.2.2.2 Zone 2: Evaporation in Boiling Liquid 78

4.2.3 Thermodynamic Trajectories for Fluid Phases 80

4.2.3.1 Thermodynamic Trajectory for Zone 1 80

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4.2.3.2 Thermodynamic Trajectory for Zone 2 82

4.2.4 Heat Transfer between Vessel Wall and Fluid Phases 84

4.2.4.1 Heat Transfer between Vessel Wall and Zone 1 86

4.2.4.2 Heat Transfer between Vessel Wall and Zone 2 914.2.5 Discharge Calculation 96

4.2.5.1 Ideal Gas Method 99

4.2.5.2 Real Fluid Method 101

4.3 MATHEMATICAL ALGORITHM FOR BLOWDOWN 106CALCULATION

4.3.1 Single Phase Algorithm

4.3.2 Two-Phase Algorithm4.4 THE COMPUTER PROGRAMME, BLOWSIM

CHAPTER 5 EVALUATION OF PERFORMANCE OF CUBICEQUATIONS OF STATE

5.1 INTRODUCTION

107111114

1165.2 PREDICTION OF THERMODYNAMIC PROPERTIES BY CUBIC 116

EQUATIONS OF STATE5.2.1 Twu-Coon-Cunningham (TCC) Cubic Equations of 120

State5.2.2 Determination of Specific Heat Capacities, Speed of 122

Sound and Thermal Expansion Coefficient from Twu-Coon-Cunningham Cubic Equations of State

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5.2.2.1 Specific Heat Capacities5.2.2.2 Speedof Sound5.2.2.3 Thermal Expansion Coefficient

123125126

5.3 EVALUATION THE PERFORMANCE OF CEOS IN 126PREDICTING FLUID DENSITIES AND PERCENTAGE OFLIQUID VOLUME FOR A MULTI-COMPONENTHYDROCARBON MIXTURE

5.3.1 Experimental Conditions and Measurement Accuracy 127

5.3.2 Results 1285.4 EVALUATION THE PERFORMANCE OF CEOS IN TERMS OF 134

PREDICTING VAPOUR AND LIQUID SPEED OF SOUND ATHIGH PRESSURES FOR SINGLE AND MULTI-COMPONENTHYDROCARBON SYSTEMS

5.4.1 Experimental Conditions and Measurement Accuracy 135for Speed of Sound

5.4.2 Determination of AAD % 137

5.4.3 Absolute Percentage Deviation as a Function of Tr and 143Pr for Vapour Mixtures

5.4.4 Absolute Percentage Deviation as a Function of Tr and 157Pr for Compressed Liquid

5.5 CONCLUSION 170

CHAPTER 6 VALIDATION OF BLOWSIM6.1 INTRODUCTION 173

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6.2 VALIDATION OF THE COMPUTERMODEL, BLOWSIM 173

6.2.1 Selection of Experimental Data 173

6.2.2 Experimental Conditions and Measurement Accuracy 176

6.2.3 Non-condensable Gas 177

6.2.4 Condensable Gas and Two-phase Mixture 181

6.2.4.1 Effect of Considering and Discounting Work Done by the 181Liquid Phase on Blowdown Simulations

6.2.4.2 Effect of Selecting Constant Heat Transfer Coefficients 202between Liquid and Wetted Wall on BlowdownSimulations

6.2.5 Effects of Assuming Ideal Gas at the Orifice on 213Blowdown Predictions and Computational Efficiency

6.3 CONCLUSIONS

CHAPTER 7 CONCLUSIONS AND SUGGESTIONS FOR FUTUREWORK

7.1 CONCLUSIONS

7.2 SUGGESTIONS FOR FUTURE WORK

214

217221

REFERENCES 223

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CHAPTER 1INTRODUCTIONThe term blowdown refers to the rapid wanted depressurisationof a vessel. In theoffshore industry for example, blowdown of vessels or sections containinghydrocarbons is a common way of reducing the failure hazard in an emergencysituation.In recent years such operations have presentedprocessand safety engineerswith adilemma.The primary aim of blowdown is to reduce the pressure and inventory in the leastamount of time possible. However, rapid depressurisation results in a dramatic dropin the fluid temperature and also tremendous heat transfer between the fluid and thevessel wall which lead to a reduction of wall temperature. If the wall temperaturefalls below the ductile-brittle transition temperature of the vessel material, rupture islikely to occur. Low fluid temperatures can also lead to the formation of solidhydrates in cases where free water is present in the vessel. The presence of solidhydrate can cause great difficulties in operations [Katz & Lee, 1990].

Clearly the optimum blowdown time requires a delicate balance between themaximum permissible blowdown duration and the minimum wall and fluidtemperatures hat may be safely accommodated.Consequently, in recent years there have been a number of theoretical andexperimental studies addressing he above issues.The theoretical models primarilyfall into two categories; those on the basis of simplified relations [API, 1990;Montgomery, 1995; Reynolds & Kays, 1958] which are either not capable ofpredicting wall and fluid temperatures accurately or are applicable to non-condensablegases only. In the majority of cases, hese models lead to gross overestimations which will in turn require considerable capital equipment expenditure.The second category are thosebasedon rigorous analytical proceduressuch as thoseproposedby Haque et al. [1992a] and Overa et al. [1994]. The main drawback in the

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former model is associated with the use of the extended principle of correspondingstates [Rowlinson & Watson, 1969] for generating fluid thermodynamic propertieswhich as well as uncertainties associated with its accuracy, makes the simulationcomputationally demanding. This also presents consistency problems as in practice,cubic equations of state (CEOS) are almost universally used in process simulations.Additionally, important information relating to the formulation of the model is notavailable in the open literature.

On the other hand, Overa et al. 1994] assumesingle phase discharge (this may beinappropriate in many practical situations) as well as a constant heat transfercoefficient for the liquid phase.The effect of the latter assumption on blowdown datahasnot been nvestigated.The purpose of the present study is to develop a mathematical simulation, namedBLOWSIM based on CEOS for blowdown of vessels containing single (vapour) ortwo-phase (vapour and liquid) hydrocarbon mixtures. The model is computationallyefficient, requires the minimum number of input parameters whilst at the same timeproduces good predictions with acceptable engineering accuracy as compared toexperimental data. In addition, various modifications to the simulation are introducedin order to identify and quantify the importance of taking into account the differentprocesses taking place during blowdown. An important part of the study involves adetailed evaluation of the performance of the recently developed TCC CEOS [Twu etal., 1992] which has been particularly designed to address some of the shortcomingsassociated with SRK [Soave, 1972] and PR CEOS [Peng & Robinson, 1976]employed in this study.

The study is divided into 7 chapters.In chapter 2, published experimental studies on blowdown reported in the past 40years are reviewed. These primarily identify the nature of the various processestaking place during blowdown.

In chapter3, themost mportantmathematicalmodels eportedn theopen iteraturefor blowdownof isolatedvessels,with no chemical eaction,undernonfire situations

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are reviewed. It starts with a description of commonly used ndustrial methodsbasedon engineering practice for blowdown calculation and is followed by introducingsimple mathematical models for blowdown of non-condensablegases.The chapter sconcluded by reviewing BLOWDOWN [Haque et al., 1992a] and SPLIT FLUIDMODEL [Overa et al., 1994] in detail for blowdown of condensablegases or two-phasemixtures.Chapter4 describes he mathematicalmodel, BLOWSIM developed in this study forvapour spaceblowdown through a single orifice from the top of an isolated vesselcontaining single (vapour) or two-phase (vapour and liquid) hydrocarbon mixtures.In the same chapter, the model is used to identify the level of detail required forblowdown modelling as well as addressing he following issues:

1) The effects of applying various cubic equations of state on the performance ofblowdown simulation in terms of predicting field data.

2) The suitability of different thermodynamic trajectories for vapour phaseexpansionduring blowdown of hydrocarbonmixtures from elevatedpressures.

3) The effects of applying different thermodynamic trajectories to the liquid phasein terms of the accuracy n predicting temperaturesandpressures.4) The effect of Overa et al. s [1994] suggestedconstantheat transfer coefficients

between liquid and wetted wall compared to the boiling heat flux empiricalcorrelation employed by Haque et al. [1992a] on the accuracyof the predictedliquid andwetted wall temperatures.

5) Comparisons of the rigorous [Haque et al., 1992a] against simple dischargecalculation methods (based on ideal gas assumption) in terms of predictingtemperaturesandpressuresduring blowdown.

In chapter 5, the Soave Redlich-Kwong, SRK [Soave, 1972], Peng-Robinson, PR,[Peng & Robinson, 1976] CEOS for generating the required thermodynamicproperties for blowdown simulation are presented and discussed. The newlydeveloped Twu-Coon-Cunningham, TCC, CEOS [Twu et al., 1992] aimed at

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addressingsome of the drawbacksof theseequations s also given. This is followedby the derivation of the appropriate equations for determining specific heatcapacities,speed of sound and thermal expansioncoefficient based on TCC CEOS.Finally the performance of the above CEOS in predicting the required importantparameters or blowdown calculation (e.g. densities and speedsof sound for vapourand liquid, as well as liquid volume percentage) for single and multi-componenthydrocarbon systems are investigated by comparison with published experimentalvalues. Mixtures containing light hydrocarbons are mainly considered as vapourspaceblowdown is always usedfor the processgas stream.In chapter 6, the results obtained from BLOWSIM are compared against thosepredicted from BLOWDOWN, and published experimental

data for high pressureblowdown of a full-size vessel containing various hydrocarbon mixtures. Theseinclude, non-condensablegas, condensablegas and two-phasemixtures. In each andevery case, the effects of incorporating any one of the three CEOS on the resultspredicted by BLOWSIM are evaluated. The latter include pressure/time andtemperature/time profiles for the vessel wall (both wet and dry), the bulk gas andwhere applicable, for the bulk liquid. Additionally, the effects of accounting for ordiscounting the work done by the liquid phase on the results obtained usingBLOWSIM are investigated. This is followed by an investigation of the effects inapplying various selectedheat transfer coefficients betweenvesselwall and the liquidphase within the vessel on BLOWSIM's predictions. The final part of the chapterinvestigates the performance of BLOWSIM basedon either ideal gas assumption orreal fluid approach at the discharge orifice in terms of minimum temperaturepredictions and computational time.Chapter 7 dealswith generalconclusionsand suggestions or future work.

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Chapter 2 Vapour Space Blowdown Phenomena

CHAPTER 2

VAPOUR SPACE BLOWDOWN PHENOMENA2.1 INTRODUCTION

In recent years, a number of experimental studies have been conducted by variousresearcherso study blowdown. Table 2.1 presentsa summary of these nvestigationscarried out in the past 40 years and the correspondingexperimental conditions. Thischapter describesthe salient featuresof these studies with a particular emphasis nhighlighting the various processes taking place during blowdown. A review ofpublished models for simulating blowdown is given in chapter3.2.2 VAPOUR SPACE BLOWDOWN

During vapour spaceblowdown of a hydrocarbon mixture, the gas within the vesselinitially expands rapidly and follows an isentropic path which leads to very low gastemperatures.However, substantial heat transfer takes place between the gas andvesselwall which prevents it from reaching the isentropic temperature.Condensationof heavier gaseoushydrocarbon components can still occur when the gas enters thetwo-phaseregion.For a vessel initially containing gas phase only, condensation will lead toaccumulation of liquid at the bottom of the vessel.The pool of liquid will be boilingvigorously because t is in contact with the relatively warm vessel wall [Haque et al.,1990; 1992b]. If the fluid is initially two-phase (vapour and liquid), liquid dropletswill be addedto the existing boiling liquid (due to reduction of pressure)within thevessel and evaporation of lighter liquid components will take place [Haque et al.,1992a]. Hence, the temperaturesof liquid and vesselwall will drop.The results of various studies [Eggers& Green, 1990; Haque et al., 1992band Overaet al., 1994] have shown that there are significant temperature differences betweendifferent fluid phasesduring blowdown. This is a clear indication of non-equilibriumbetween phases. The temperature differences also cause non-uniform walltemperatures along the vessel. The differences in vessel wall

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Chapter 2

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Chapter 2 Vapour SpaceBlowdown Phenomena

and fluid temperaturesare maintained for a significant period of time [Haque et al.,1992b; Overa et al., 1994]. The following describes he main featuresof heat transfereffects between vessel wall and vapour space,vessel wall and liquid space, vesseland surrounding, heat and masstransfer between fluid phases and flow through therelief valve during blowdown.2.2.1 Heat Transfer betweenVesselWall and Vapour SpaceBlowdown of non-condensablegaseshave been studied for a number of years asindicated in table 2.1 which also gives the pertaining experimental conditions.Reynolds and Kays [1958] for example analysed experimentally the discharge of asmall air tank for a short time (ca. 20 s) by measuring the bulk gas temperatures.Their results indicated that in practice, the reduction in the bulk gas temperaturesdoesnot follow the isentropic path concluding that heat transfer between the gas andwall is significant. The authors developed a method for predicting gas temperaturesby assuming natural convection taking place in the vessel. The effect of forced.convection due to discharged gas was ignored. Calculated gas temperaturesagreedwell with the experimentalvalues.Potter and Levy [1961] found that during depressurisationof moist air cylinders, thegas temperatures deviated from isentropic temperatures and passed through aminimum followed by an increase near the end of the tests. Byrnes et al. [1964]conducted experiments using a small hydrogen tank for different blowdown times(see able 2.1). The eventual recovery in the temperatureof the gaswas considered obe due to the increasingly pronounce,ffect of heat transferbetweenthe gasand vesselwall. On the bases of Reynolds and Kays' [1958] conclusion regarding naturalconvection, the authors developed a method for estimating gas temperatureswhichwas capable of providing a fair agreementbetween calculated and measuredvalues.These studieswere mainly restricted to small scalenon-hydrocarbon fluids in whichnatural convection was the main mode of heat transfer as opposed to forcedconvection. The wall temperatureof hydrogen tank during blowdown was measuredby Bynes et al.. In all cases,the wall temperature did not drop as low as the gas

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temperatures.This was mainly attributed to the low gasheat transfer coefficient andthe relatively high heat capacity of the vesselwall.More recently, relatively high pressureblowdown experimentshave been conductedfor both hydrocarbonsand non-hydrocarbonsusing different scaleplant. Haque et al.[1990,1992b] for example presentedexperimental data for depressurisinga smallvesselcontaining either pure nitrogen or its mixture with carbon dioxide (70 mole %nitrogen, 30 mole % carbon dioxide) and full scalevesselscontaining hydrocarbongas mixtures at high pressures (see table 2.1 for initial operating conditions).Nitrogen was used asa non-condensablegaswhilst the nitrogen and hydrocarbon gasmixtures represented a condensable gas. The fluid and wall temperatures weremeasuredat various positions along the vesselusing 76 thermocouples for the smallvesseland 156thermocouples or the full scale vessel.The variations of measuredbulk fluid and wall temperatureswith time for nitrogenand the hydrocarbon gas mixture are shown in figures 2.1 - 2.3. The grey regionsshow experimental measurementswhilst the solid lines show the predictions fromBLOWDOWN [Haque et al., 1992a]. The pressure/timeprofile for the hydrocarbongas mixture is shown in figure 2.4. Figure 2.5 shows the isotherms and flow patternduring blowdown of nitrogen.Overa et al. [1994] presentedexperimentaldata for depressurisinga vesselcontaininga hydrocarbon gas and unstablised oil (composition was not specified). Thevariations of measuredbulk fluid temperatureswith time is shown in figure 2.6. Thesolid lines represent predictions from a computer program based on the SPLITFLUID MODEL developed by the sameauthors.The initial operating conditions aregiven in table 2.1.Referring to figures 2.1 and 2.2, the grey regions representing he measuredbulk gastemperaturesare a clear indication of presenceof temperaturegradientswhich in turngive rise to density differences within the vapour phaseespecially after an extendedperiod of time. Figure 2.5 on the other hand, indicates the presence of largetemperature gradients near the walls of the vessel which lead to significant naturalconvection within the vessel. The sameobservation was also made by Overa et al.

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[1994] who measured gas temperaturesduring blowdown at different elevationsalong the vessel and filmed the outside of the vesselwith a heat sensitive film. Theauthors were able to map the fluid flow pattern for the vapour space duringblowdown indicating the prevalenceof natural convection for a variety of blowdowntests. Figure 2.7 shows the generalpattern. These flow patterns are very similar tothosegeneratedby Haque et al. [1992b] as shown in figure 2.5.

Overa et al. [1994] interpretedthe presenceof temperaturegradients in vapour phaseduring blowdown as a result of warming effect of cold vapour by the warmer vesselwall and accumulation of colder vapour in the bottom of the vessel. Haque et al.[1992a] also pointed out that natural convection is pronounced especially at highpressures due to low viscous effectswhich facilitate natural convection. Thepredominant effect of natural convection was also observedby Overa et al., [1994]during experiments with vesselscontaining substantially warmer liquid where coldgas would accumulate mmediately above he liquid.Returning to figures 2.1 and 2.2, the bulk gastemperaturespassthrough a minimumand rise near the end of blowdown. This hasbarn,explained by the competition of twoprocesses;gas expansion cooling and convection heating of the gas by vessel wall.Referring to figure 2.4, at the beginning of blowdown (before 500 s), the rate ofdepressurisation is so high that the gas temperature drops very rapidly due toexpansion as comparedto relatively slow processof natural convection. After 500 s(figure 2.4), the depressurisationrate slows down and the effect of heat transferbetweengasand vessel wall dominates he gas expansioncooling effects. As a result,the gastemperaturerises.Norris [1993,1994] also made the same observation during the blowdown of highpressure vessels containing air or a hydrocarbon gas mixture. The experimentalconditions can be found in table 2.1. Samemechanismsdescribedabovecan be usedto explain the observedbulk gastemperatureprofile shown in figure 2.6 by Overa etal. [1994]. Interestingly, the data indicate two local minima instead of one in themeasuredbulk gas temperatures.The authors attributed this to seven urbulence in

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Chapter 2- Vapour Space Blowdown Phenomena

vapour phase and possibly liquid entrainment on the thermocouples which led toerratic vapour temperature measurements.The temperatureof the wall in contact with vapour as shown in figures 2.1 and2.3 donot drop as low as that of the gas. This is mainly due to low gas heat transfercoefficient and the high heat capacity of the vessel wall. However, from figure 2.3,the lowest wall temperature s located n the regions in contact with the liquid phase.

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Chapter 2

300

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Y

Time /s

Figure 2.2 Variations of bulk gas and bulk liquid temperatures with time fordepressurisation of a hydrocarbon mixture containing 66.5%mole methane, 3.5 mole /, ethane, 30.0 mole % propane andtraces of higher molecular weight hydrocarbons in particular ofbutanes (Hatched regions span experimental measurements, solidlines are predictions from BLOWDOWN) [Haque et al. 1992b[

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Chapter 2

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120

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Figure 2.4 Variation of pressure with time for depressurisation of ahydrocarbon mixture containing 66.5% mole methane, 3.5 mole% ethane, 30.0 mole % propane and traces of higher molecularweight hydrocarbons in particular of butanes (Solid line isprediction from BLOWDOWN) ]Haque et al. 1992b]

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Figure 2.5 Isotherms (left-hand side) and flow pattern (right-hand side)during blowdown of nitrogen IHaque et al., 1992b1

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eaI

0 2 4 6 8 10 12 14Time(min)

16 18 20 22

Figure 2.6 Variations of bulk gas and bulk liquid temperatures with time fordepressurisation of a hydrocarbon two-phase mixture 10vera etal., 19941

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Figure 2.7 Flow pattern of gas phase in a vertical vessel during blowdownlOvera et at., 19941

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2.2.2 Heat Transfer between VesselWall and Liquid SpaceEggers and Green [1990] presented results of depressurising a small vesselcontaining 86 vol. % of liquid carbon dioxide (see table 2.1 for experimentalconditions). The location of the thermoelements and the corresponding recordedtemperatures/timeprofiles at different positions along the tank are shown in figures2.8 and 2.9 respectively.Also shownin figure 2.9 is the pressure/timeprofile.Figure 2.9 indicates that before formation of dry ice (where the pressure remainsconstant) the liquid temperature (curve T6 ) is very similar to the temperature of theinner wall by the liquid (curve T11) at the bottom of the tank. The similarity of liquidand inner wall temperatures indicates good heat transfer between the liquid andvessel wall. The same observation may be made from the data in figures 2.2 and 2.3.

Haque et al. [1992a] explained the above behaviour by attributing it to nucleate,transition and film boiling of the liquid phase which yields high heat transfercoefficients when compared with natural convection in gas phase [Welty et al.,1984]. Therefore, as shown in figure 2.2, the bulk liquid temperaturesare initiallyhigher than the bulk gas. During the latter stages of blowdown, due to the rise ofvapour temperatureand alsobecausemost of the heat from the vesselwall in contactwith the liquid is removed, the liquid temperature drops below that of the gas.Consequently, the vessel wall temperaturesas shown in figure 2.3 are lower at thebottom of the vessel thanat the top. Therefore, it is common to find the minimumwall temperature located at the bottom part of the vessel where liquid (either fromcondensationof gas or existing liquid) is present.Note, incidentally, that, while thereis some spatial variation in bulk gas temperature, here is very little spatial variationin bulk liquid temperature.This is because he intenseboiling in the liquid gives riseto very rapid mixing andhence hermal equilibration.Another interesting observationby Haque et al. [1990] relating to the blowdown of acondensablegas is a rise followed by a decreaseof condensate emperature. Suchdata for a nitrogen and carbon dioxide mixture is shown in figure 2.10. The upperband refers to bottom zone (condensate) emperaturesand the lower band to top zone(vapour) temperatures.The corresponding predictions using BLOWDOWN are also

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shown. The liquid level is only a few centimetres deep compared to the vessel heightof 1.5 m. The authors attributed this behaviour to the warming effect of the relativelywarm vessel wall in contact with the small amount of condensate formed. The morevolatile components were driven off and thereby raising the boiling temperature.During the latter stage of depressurisation, due to expansion, more and morecondensate was formed in the upper part of the vessel. This fell to the bottom and asthe bottom of the vessel itself was cooled, the evaporation rate fell off and a pool ofliquid gradually accumulated. Since the pressure in the system was still falling, theliquid experienced evaporative cooling. It is interesting to note that the suddenincrease of liquid temperatures is not found in figure 2.2 where the condensate levelwas reported to be appreciably higher compared to the vessel height.

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Figure 2.8 Schedule of thermoclements positions of tank containing carbondioxide Eggers & Green, 19901

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0 120 240 360 480Time/s 60("

Figure 2.9 Time profiles of pressure and fluid and wall temperatures(blowdown time is 400 s) of a carbon dioxide tank [Eggers &Green, 19901

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Chapter 2 Vapour Space B[owdown Phenomena

deVCLEV

0 20 40 60Timels

so 100

Figure 2.10 Variations of bulk gas and bulk liquid (condensate) temperatureswith time for depressurisation of a vessel containing 70% molenitrogen and 30% mole carbon dioxide (Hatched regions spanexperimental measurements, solid and dotted lines arepredictions from BLOWDOWN) [Haque et al., 19901

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2.2.3 Heat and Mass Transfer Between Vapour and Liquid PhasesDuring blowdown of condensablegases or two phase mixtures, heat and masstransfer take place by evaporation and condensation due to drop in pressure andtemperature.Overa et al. [1994] also consideredheat transfer by convection due totemperature difference between phases. Mayinger [1982] indicated when thedepressurisationrate is high during blowdown, thus giving rise to transient unstableconditions, boiling and condensation delays exist in liquid and vapour phasesrespectively. Additionally, the processof phaseseparation of evaporated iquid andcondensedvapour from corresponding phasescan be relatively slow when comparedwith high rate of depressurisation.The author demonstrated he effects of delay and phaseseparation n boiling liquidby depressurisinga vessel2/3 filled with saturated iquid, refrigerant R 12, from 7.4atm and 300C to ambient pressurewithin 15 s. The variations of pressure with timeand fluid temperatureare shown in figure 2.11.Referring to curve A, depressurisation starts at point B. During the very steeppressuregradient betweenpoints B and C, the measured iquid temperaturemarkedlyexceeds the saturation temperature (see curve B), which indicates considerableboiling delay. At point C, bubble formation starts n the liquid. Due to relatively slowprocessof phase separation, he dispersion level moves upwards during the period C-D and reachesthe releasevalve. The vapour flow at releasevalve containing onlytraces of liquid droplets is supersededwith a two-phase discharge containing largeamounts of liquid. As the maximum velocity of two-phase mixture is much lowerthan sonic velocity of vapour, vapour formation in the vesselexceeds he volumetricdischargerate between time D and F. As a result, pressure starts to build up withinthe vessel until point F where the rate of flashing starts to fall and the pressuredecreasessteadily to point H.

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2.2.4 Heat Transfer Between the Vesseland the SurroundingThe type of heat transfer between the vessel and the surrounding depends on thenature of the surrounding atmosphere.Under non-fire situations, heat transfer is bynatural convection if the vessel s sheltered, or example, within an enclosed moduleon an offshore platform and the wind speed s low. Otherwise, heat transfer may beby forced convection.2.2.5 Flow Through the Relief ValveIn the caseof a fluid (single or two phase) approaching a relief valve, if the back-pressure s sufficiently low, the fluid will be acceleratingthrough the orifice at it'smaximum velocity. In addition, condensation may occur due to rapid rate ofexpansion from the upstream to orifice pressure.Haque et al. [1990] indicated thatthe fluid in the choke could be either in a metastablestate or in thermodynamic andphase equilibrium. The authors compared predictions based on the aboveassumptions with experimental measurements and concluded that the latterassumptionsgavebetter results.

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Chapter 2

p(badAPo


Curve A

liquid30oc

0B\ _anl-! r \ . i. 5oaraes-54

. v-Curve B

i /_\ ''_ _; _Sn 6

P

0 rw

Figure 2.11 Variations of pressure with time (curve A) and fluid temperaturewith pressure (curve B) for depressurisation of saturated liquid,refrigerant R 12 [Mayinger, 19821

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2.3 CONCLUSION

On the basis of the literature reviewed so far, we can make the following importantconclusionswith regards o the various processesaking place during blowdown:

" The gas temperaturedoesnot drop as low as the isentropic temperature.After anextended period of time the gas temperature reaches a minimum and increasesnear the end of blowdown. This is due to heat transfer between gas and vesselwall.

" The presenceof temperaturegradientswithin the gas phase gives rise to densitydifferenceswhich in turn lead to natural convection dominating forced convectionasthe main heat transfermechanism.

" The temperatureof vesselwall in contact with the gasdoesnot drop as low asthegastemperaturebecauseof low gasheattransfer coefficient andhigh heat capacityof the vesselwall.

" The liquid phase is superheatedand boiling occurs. This results in a high heattransfer coefficient when compared o natural convection in gasphase.

" Due to rapid rate of heat transfer betweenvessel wall and liquid, it is common tofind the minimum wall temperature ocatednear the bottom of the vessel.

" There are significant temperature differences between different fluid phasesduring blowdown. Hence,equilibrium doesnot exist betweenthesephases.

" Inter-phase mass and heat transfer are achieved by evaporation of lighter liquidhydrocarbon components andcondensationof heavier gaseoushydrocarbons.

" During rapid depressurisation,boiling and condensationcan be significant. Phaseseparationof evaporated iquid and condensedvapour from corresponding phasescan in turn be slow when compared with the high rate of depressurisation.Theabove may lead to foaming and two-phase discharge and a build up of pressurewithin the vesselduring blowdown.

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" The flow (either gas or two-phase) through the dischargeorifice is usually choked(at the speedof sound for gasor at it's maximum for two-phasedischarge) if theback-pressure is sufficiently low. The fluid in the choke may in turn be in ametastablestate or in thermodynamic and phaseequilibrium.

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Chapter 3 Reviewof Mathematical Models for Blowdown Simulation

CHAPTER 3

REVIEW OF MATHEMATICAL MODELS FOR BLOWDOWNSIMULATION3.1 INTRODUCTION

This chapter reviews some of the most important mathematical models reported inthe open literature for blowdown of isolated vessels, with no chemical reaction,under non fire situations. It starts with a description of commonly used industrialmethods based on engineering practice and is followed by introducing simplemathematical models for blowdown of non-condensable gases. The chapter isconcludedby reviewing mathematicalmodels for blowdown of condensablegasesortwo-phase mixtures. In caseswhere experiments were carried out for validation ofthe reviewed models, the pertaining experimental conditions including the volume ofvessel and measuredparametersmay be found in table 2.1 (chapter2).3.2 TRADITIONAL ENGINEERING METHODS FOR BLOWDOWN SIMULATION

A number of simple methods for blowdown are in common use in industry. Theseuniversally assume hat the fluid within the vessel is homogenous.The selection ofthermodynamic path for expansion of fluid that takesplace during blowdown is oftenarbitrary. Overa et al. [1994] studied some traditional engineering methods given intable 3.1 and compared heir predictions with experimentaldata.Table 3.1 Traditional blowdown methods used in industryThermodynamic Path Description100% or 50% The fluid expansion is either 100% or 50% isentropic (i.e., firstisentropic isentropic, then isenthalpic with the enthalpy corrected by an efficiency)

and there is no heat transfer with the vessel.Isenthalpic The fluid expansion is isenthalpic. No heat transfer with the vessel is

assumed o take place.Constantheat ransfer The fluid expansions 100% sentropic,andheattransferbetween luidcoefficient and vessel wall is determined by assuming constant heat transfer

coefficientswhich are specifiedseparatelyor vapourand iquid phases.

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Chapter 3 Reviewof Mathematical Modelsfor Blowdown Simulation

The authors compared predictions of minimum fluid and wall temperatureswithblowdown experimental data of low pressure air (initially at 21.2 atm) and highpressurenitrogen (initially at 175.7 atm). For a low pressure wo-phasehydrocarbonmixture (initially at 19.7atm), comparisonswere only made for the minimum vapourand liquid temperatureswhile the minimum wetted and unwetted wall temperatureswere not reported.Their results indicated that the predictions from the abovemethods greatly deviatedfrom experimental values except for low pressure air, where the constant heattransfer method gave reasonablepredictions of both gas and wall temperatures.Thelarge errors introduced by all the above models were attributed

to unrealisticassumptionsof the mode of heat transfer within the vessel and also to the modellingof a two phasemixture as a single homogenous luid.

As mentioned in chapter 2, the heat transfer effects of the fluid and non-equilibriumbetween phasesare the major features of blowdown. Hence, correct interpretationsare required for such effects in order to generateaccuratepredictions.3.3 SIMPLE MATHEMATICAL MODELS FOR BLOWDOWN OF NON-CONDENSABLE

GASES

It is clear from the above that the application of a proper energy balance betweenvessel wall and the fluid within the vessel is required for blowdown calculation. Thesimplest case is blowdown of a non-condensablegas where the effects due to thepresenceof vapour condensateor liquid phase can be ignored. The following is areview of the pertaining mathematical models reportedby various authors.These aremainly derived from the first law of thermodynamics. Heat transfer from thesurrounding to the vessel s ignored.3.3.1 Reynolds and Kays [1958]Reynolds and Kays [1958] proposeda mathematicalmodel which allows predictionsof gastemperatureand dischargerate as a function of residual mass of gas n a vesselwith or without solid 'capacitors. In some industrial applications, capacitors are' In someapplications,heat capacitorsaredeliberately inserted n the container to control temperatureor storethermal energy.

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deliberately placed within the vessel to control temperatureduring blowdown. Heattransferbetweengas and the vesselor the solid capacitors s accounted or.The major assumptionsadopted n the simulation are :1) Constantspecific heat for vesselwall.2) Constant heat transfer coefficient between the gas, vesselwall and internal solid

capacitor.3) Temperaturegradient across he vessel wall or solid capacitor is ignored.4) The gas n the vessel s ideal andwell mixed.

The authors derived an energy balance in terms of dimensionless gas temperature,T*, and a dimensionlessmassof gas,M* in the vessel n the form :

' 3.3.1W"M" dT. -[(k-1)w*+NTU]T" +NTU T, = 0dM

The dimensionlessparametersare:M* _NTU =

M/Mp(hA)/(CvWo)

T* = T/TpTC* = TcTow* = w/wo

Where A= Heat transfer areah= Heat transfer coefficient between fluid and vessel wall or

solid capacitorCV = Gasspecific heat at constant volumek= Ratio of specific heatsof the gasM, MO = Residual mass of gas in the vessel at any instance during

blowdown andat initial condition respectivelyT, To = Gastemperatures t any instanceduring blowdown and at

initial condition espectivelyTc = Vessel wall's or capacitor's temperatures at any instance

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Chapter 3 ReviewofMathematical Modelsfor Blowdown Simulation

w, w0during blowdownDischarge rates at any instance during blowdown and atinitial condition respectively

The initial conditions when M* =I are :T =1Tc = Tco

Where Tco* is equal to Tao/To, and Tcotemperatureat initial condition.

(3.3.2)(3.3.3)

is the vessel wall's or capacitor's

Equation 3.3.1 is based on the first law of thermodynamics for a non-steady flowprocess. During blowdown through a choke, the discharge rate equals to the criticalflow and the dimensionless temperatures of vessel wall or capacitor are assumed tobe constant. This assumption is said to be appropriate when blowdown time is short.The dimensionless discharge rate, w*, given by the authors is :

w* = M*(T*)y2Hence,equation3.3.1 is reduced o :

" T` T NTUM* dT `_ (k -1T` +=0dM M`T`IV2

(3.3.4)

(3.3.5)

The dimensionlessvessel pressurebasedon ideal gas equation is:

P* = M*T*Where P* =

P,Pp =

(3.3.6)

P/POVessel pressures at any instance during blowdown and atinitial condition respectively

As equation3.3.5 s nonlinear, t canbe integratednumericallyeitherby specifyingdT* or dM*. w* and P* are then calculatedfrom equations3.3.4 and 3.3.6

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respectively. The actual temperature,pressureand massof gas in the vessel can bedeterminedprovided the initial conditions areknown.In order to investigate the mode of heat transfer during blowdown and to validatetheir model, the authors performed experiments by depressurising a low pressurevessel containing air from an initial pressureof 7.8 atm through a choke. A typicalrun lasted for about 20 s. Two types of thermal capacitors, vertical-strip andconcentric-can were inserted into the vessel respectively. The pressures and gastemperatureswere recorded for both cases.The vessel was lined internally with athick layer of balsawood to provide an adiabaticenvironment.For any measuredpressureand gastemperature, he dimensionlessgas mass,M* wasdetermined from equation 3.3.6. The results of T* against M* for a vertical-stripcapacitor and a concentric-can capacitor aregiven in figures 3.1 and 3.2 respectively.The solid lines in the figures represent the predicted values while the data pointsshow the measured values for two different runs. The dotted lines on the other handare predictions on the basis of the adiabatic expansion of gas which show largedeviations from experimental data.By adjusting the values of NTU in equation 3.3.5, the authors were able to producegood predictions and also determine the heat transfer coefficients for both types ofcapacitors. The magnitude of calculated heat transfer coefficients suggested thedominanceof natural convection in the gasphase.The authorsalso indicated that theGrashof number of the gas was greater than 109and turbulent convection boundarylayer existed in the gasduring blowdown.

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Chapter 3

L00

o9S"T'040

aes

o. o

Reviewof Mathematical Modelsfor Blowdown Simulation

CRITICALLOWBLAWOOWNV=1.00NTU= 2.00

RUN- " \6.rUN ..AOIABATIC- \\.SOLUTION

1.o 035 0.90 o. s o. o o.sM" 0.70 0.63 Q60'Figure 3.1

1.00

0.95T'oso

045

o. a

Variation of dimensionless temperature, T*, with dimensionlessmass of gas, M* for blowdown of high pressure air with vertical -strip capacitors [Reynolds & Kays, 1958]

CRITICALLOWSLOWDOWNT* =1.00NTU - 4.00

0-RUN. 70- RUN 8 0ADIABATICSOLUTION

LOO o9s 0.90 O. 5 '' OAO O.7 . 0.70 - Q6M" 0.60

Figure 3.2 Variation of dimensionless temperature, T*, with dimensionlessmass of gas, M* for blowdown of high pressure air withconcentric -can capacitors [Reynolds & Kays, 19581

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3.3.2 Byrnes et al. [1964]The dominance of heat transfer by natural convection also at high pressureswasconfirmed by Byrnes et al. [1964]. The authors useda correlation given by McAdams[1954] for predicting the heat transfer coefficient between the gas and vessel wallduring natural convection to produce good agreementwith experimentaldata.Similar to Reynolds andKays, Byrnes et al. assumedhat the gaswithin the vessel obe ideal. On the basis of the first law of thermodynamics for steadyflow process, heenergy balance between the vessel wall and the gas was expressedas a function oftime instead of the dimensionlessparametersused by Reynolds and Kays. This isgiven by:

hA(TW - T) =M Cp dTdt -v dPdt(3.3.7)

The pressure-timeprofile is given in an exponential form provided the flow is chokedacross he orifice:

P= CatPo

Where a= Pressureprofile parameter(fitting constant)Cp = Gasspecific heat at constantpressureV= Vessel's volumeTW = Vesselwall temperaturet= Time

(3.3.8)

If the pressure profile parameter, a, in equation 3.3.8 is known, equation 3.3.7 canthen be solved by finite difference method where the mass of gas in the vessel,M, isassumedconstant for a given time interval. The amount of dischargedgas as a resultof the expansion s given by the volume of fluid after expansionminus the volume ofthe vessel.Material balance s then performed to correct the value of `M' for the nexttime interval.

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For validation purposes,the authors carried out experimentsby depressurising highpressure hydrogen vessels (initial pressure is 135.2 atm) with three differentblowdown times, 14,30, and 480 s. The pressure,gas and wall temperaturesweremeasured.Figures 3.3 - 3.5 show the comparisonsbetween experimental data andpredictions from the model (solid lines represent measured values, dotted linesrepresentpredicted values) for the three different blowdown times respectively. Thepredictions are based on the assumption of constant wall temperature. Reasonableagreement s observed rom the abovefigures.Byrnes et al. [1964] demonstrated he applicability of the perfect gas assumptionforblowdown simulation of a simple gas at elevated pressures. Both mathematicalmodels described above are simple to use and give reasonable gas temperaturepredictions. However, in the caseof vesselscontaining gasesat low temperaturesandalso when the vessel is depressurised or a significant period of time, Reynolds andKays [1958] assumption of constant vessel wall temperaturewill fail to predict thelow vessel wall temperatureswhich may be encountered n practice. On the otherhand, although Byrnes et al. s [1964] method accounts for the variation of heattransfer coefficient with time, it is not completely predictive as the pressure-timeprofile is needed rom experiment.

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Chapter 3 Review of Mathematical Models for Blowdown Simulation

2000

1500LA-

GAS TEMPERATUEXPERIMENTAL

1000

500

CALCULATED5 10TIME, SEC.

:5NCLWNwaL

Figure 3.3 Variations of temperatures and pressure for blowdown of highpressure hydrogen (blowdown time = 14 sec) [Byrnes et al., 19641

A12000

_W.1 00=

800dPRESSURE00400'

ICALCULATED 200MR..XPERIMENTALI '010 20 30TIME,SECFigure 3.4 Variations of temperatures and pressure for blowdown ofhigh

pressure hydrogen (blowdown time = 30 sec) [Byrnes et al., 19641

AVERAGEALL

14

-18001600

TEMPERATURE-I 14001200

0

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Chapter 3 Review of Mathematical Modelsfor Blowdown Simulation

7066

4-545862

IN \I!50A N1ooo%

AVERAGEALLTEMPERATURE1500

-- , 54 .. w4642 SYSTEM- ` PRESSURE500

26

33

38 \\ GAS EMJAS EMP.(EXPERIMENTAL) Z/GAS TEMP.(CALCUTATED)

I1 23456TIME,MIN 78

500

Figure 3.5 Variations of temperatures and pressure for blowdown of highpressure hydrogen (blowdown time = 480 sec) [Byrnes et al., 19641

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3.3.3 Montgomery [1995]The abovedrawbackswere addressedby Montgomery [1995]. The author relatedtherate of energy loss in the vesselwall to the rate of energy absorbedby the fluid. Theenergy balance, which was based on the first law of thermodynamics of unsteadyflow process, was simplified by assuming constant gas specific heat capacity andconstantheat transfer coefficient betweenthe gasand vesselwall. The correspondingpressure,wall and gas temperatures ogether with gas compressibility factor werethen determinedby solving the energyandmassbalances teratively.

The material and energybalanceswere basedon a vesselwith an arbitrary number ofinlet and outlet streams asdepicted n figure 3.6.

Inlets Outlets

0

Figure 3.6 A pressurised vesselwith a number of inlet and outlet streams[Montgomery, 1995]

The material balance s given by :d(pV) m1

dt_ 1(Idt

J iWhere p= Fluid density within the vessel

m= Mass of fluid= Mass lowrateof stream in or out of thevesselCdt)

(3.3.9)

The rate of changeof fluid energy is given by summation of three components.Thefirst component representsthe energy transfer of the vessel due to inlet and outletmass flows. The second component is the work done by the expanding fluid in thevessel. The third component is the energy transferred from the vessel wall to thefluid. The energybalance for the fluid is given by :

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d(pVCPT)-

(d(mCpT))+ dPV) + hA(TW - T)dt dt dt

and :

i

hA(TW - T)d(MWCpWTW)

=- dtWhere CpW

Mwd(mCpT)

dt i

(3.3.10)

(3.3.11)

Specific heat capacity of the vessel wall material atconstant pressureMass of the vessel wallEnthalpy flowrate of stream i in or out of the vessel

For flow into the vessel, the specific heat, Cp, and fluid temperature,T, of stream iaredeterminedbasedon the conditions of inflowing fluid. For flow out of the vessel,both parametersarebasedon the conditions of fluid within the vessel.The equation to determine flowrates in and out of the vessel s given by the author inthe form :

dm = 0.525Yd2 PK

and

dt

fLD

(3.3.12)

(3.3.13)

Where d= Internal diameter of pipe, valve or fitting (in.)D= Internal diameterof pipe, valve or fitting (ft.)f= Moody friction factorL= Length of pipe or equivalent length of valve or fitting (ft)Y= Expansion factor

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AP = Pressuredrop acrosspipe, valve or fitting (psia)The expansionfactor, Y, is given by :

Y=- 0,515 +0.291 aK O. 47 P1

Where P1the inlet pressure o pipe, valve or fitting in psia.

(3.3.14)

The author then investigatedthe variation of gasand vessel wall temperaturesduringblowdown for different heat transfer coefficients (0 - 500 btulhr-ft2) by applying themodel to a hypothetical high pressure initial pressureand temperatureare 103.1atmand 311 K respectively) full-size vessel (internal diameter and vessel height are 1.5m and 4.8 m respectively). The gas was assumed o be ideal with a molecular weightof 20g/mol. The results of variations of gas and vesselwall temperatureswith timefor different heat transfer coefficients aregiven by figures 3.7 - 3.10.Two major conclusions were drawn at the based on these studies. Firstly, asexpected,the gas temperaturenever reaches he isentropic temperature (from figure3.7, -175F) during blowdown. Secondly, the vessel wall temperaturedoes not dropas ow asthe gas.Although no experimentaldatawere usedto validate the model, theconclusions confirmed the experimental observations described in section 2.2.1 ofchapter 2. The author also indicated that the effect of liquid vaporization could besignificant. Hence, more sophisticated mathematical models are required to handlethe additional effects due to presenceof liquid or condensedgas.

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Chapter 3

100u_.:0 50. '=..Q- -50v.. -ioo, .._._.Y-150

:: _240

Review of Mathematical Modelsfor Blowdown Simulation

eratureiG.

emps { .{. ... . . I ...:. _.... _ ...

. ..:...F. I

LI l

0 20 40. .. 60Time,sec _80 100 - 120

Figure 3.7 Variation of temperature with time for gas with no heattransfer with vessel(h =0 Btulhr-ft2 F) [Montgomery,19951

100u. 800 60

A 40.: W i".. t". -i .-._Q _.20'H 0

-20

Vesselsteel temperature

Gasemperature..

0 20 40 ' 60 ,. . 80: 100 * 120Time,secFigure 3.8 Variations of gas and vessel wall temperatures with time

for gaswith heat transfer with vessel (h = 50 Btu/hr-ft2 F)[Montgomery, 1995]

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Vesselteelemperature

0 20 40 60 80Time,sec 100. 120

Figure 3.9 Variations of gas and vesselwall temperatures with timefor gas with heat transfer with vessel (h = 100 Btu/hr-ft2-F) [Montgomery, 1995]

100 80

60"L t'; ac tamnarattirra....... ': ' ....... . v... f. v.40 ig' 20

0 _i 0 20 40 60Time,sec80 1- 100 120

Figure 3.10 Variations of gas and vesselwall temperatures with timefor gas with heat transfer with vessel (h = 500 Btu/hr-ft2-F) [Montgomery, 19951

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3.4 RIGOROUS MATHEMATICAL MODELS FOR BLOWDOWN

So far, two models, BLOWDOWN [Haque et al., 1990; Haque et al., 1992a] andSPLIT FLUID MODEL [Overa et al., 1994], have been reported for determination ofpressure and temperature time profiles during blowdown of a vessel containing ahydrocarbon mixture. These take into account non-equilibrium effects as well astemperature differences between phases and their associated vessel wall. Bothmodels have been validated against experimental data [Haque et al., 1992b; Overa etal., 1994]. Their main features are described in the following.

3.4.1 Blowdown [Haque et a1.,1990; Haque et al., 1992a)The mathematicalmodel, BLOWDOWN, developedby Haque et al. [1990,1992a] atImperial College is aimed to provide accuratepredictions of physically significanteffects taking place during blowdown. It is in two versions; the earlier version[Haque et al., 1990]canhandle two phasesmixtures while the later version [Haque etal., 1992a] s capableof handling an additional third phase,water. The earlier versionof BLOWDOWN is describedbriefly here while the more superior revised version,which has been validated with high pressurehydrocarbon mixtures, is reviewed inmore detail.3.4.1.1 Haque et a1.,1990The first version of BLOWDOWN [Haque et al., 1990] was developed to handlevapour spaceblowdown (blowdown from vessel's top) of permanent or condensablegases.The depressurisationprocess s approximated by dividing it into a seriesof discretepressuresteps.The vesselduring blowdown is assumed o be divided into two zones:Zonel: Vapour togetherwith any suspendediquid-phase droplets, below which isZone2: Condensedvapour from the top zone forming a pool on the bottom of the

vessel.This zone is eliminated if liquid is not present.Spatially uniform temperatureand composition are assumedn eachzone whilespatially uniform pressures assumedwithin thevessel.For eachpressurestep, he

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fluid is assumed o expand sentropically followed by heat transfer from the adjacentvessel wall.The mathematical algorithm adopted s given in the following :1) Selecta pressuredecrement.2) Perform an isentropic flash on each zone.3) Calculate the rate of discharge hrough the choke.4) Calculate the duration of the time step andthe amountof fluid discharged.5) Calculate the heat transfer coefficients for eachzone.6) Perform energy and massbalancesover the contents of each zone and an energy

balanceover the vesselwall.7) If depressurisation is complete, stop; otherwise repeat this process.

On the basesof experimental observation (chapter 2), natural and forced convectionbetween Zone 1 and adjacent vesselwall are considered.While in Zone 2, nucleateand film boiling heat transfer involving higher heat transfer coefficients comparedtoZone 1 areassumed.The main mode of heat transfer betweenvesseland surroundingis assumed o be natural convection. The corresponding correlations for predictingheat transfer coefficients and details of dischargecalculation will be described laterin the revised version of BLOWDOWN (section 3.4.1.2)The authors validated the model against experimental data obtained followingblowdown of a small high pressurevessel (initial pressure is 148 atm) containingeither pure nitrogen or a mixture of nitrogen and carbon dioxide. The results of thecomparison for nitrogen mixture were shown earlier in figure 2.10 of chapter2.In the case of nitrogen blowdown, the model successfullypredicted the temperaturesof gas and of the inner wall. While for the mixture of nitrogen and carbon dioxide,the mathematical model was able to predict when condensationoccurred and also thecorrespondingtemperaturesof vapour and condensate.

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3.4.1.2 Haque et A, 1992aThe revised version of BLOWDOWN incorporatestwo major modifications. Firstly,the isentropic expansion of the fluid is replacedby a polytropic process [Bett et al.,1975] as it allows considerably larger pressuredecrementswhile retaining accuracy[Haque et al., 1992a]. Secondly, the model can handle an additional third phase,water, which is commonly found in offshore unit operations.The vessel is divided into three zonesprior to blowdown as shown in figure 3.11.Theseare :

Zonel: Gaseoushydrocarbon including evaporatedwater, below which isZone2: Liquid hydrocarbon including dissolvedwater, below which isZone3: Freewater including dissolvedhydrocarbonsMATHEMATICAL ALGORITHM

A polytropic process Bett et al., 1975] is defined as hat in which both heat and workare transferred.By assumingthe process s reversible, it can be used to approximatean irreversible process involving a real fluid when the initial and final statesof theprocessareknown. Bett et al. [1975] indicated that such approximation usually givesremarkably accuratepredictions of heat and work.There are an infinite number of reversible polytropic paths between two states.Bettet al. proposed the following path which is physically plausible and mathematicallyconvenient :

,(dTJ -CWhere C= Polytropic constant

S= Fluid entropyT= Fluid temperature

(3.4.1)

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/100Zone I: Gas

Zone 2: Liquidhydrocarbon

Zone 3: Freewater

I\

To flareor vent

Choke

Figure 3.11 Schematic diagram of vessel with blowdown from topI1laque et al., 1992x]

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The above equation is solved by assuming a linear relationship between heat andtemperature.The amount of heat being transferred s given by :

Q= 2TdS= C(T2- Tl)

Where Q= Heatbeing transferred1,2 = Initial and final states

Hence,

n= (S2 -S1)(T2 -TOl< inlT2 T1J

(3.4.2)

(3.4.3)

and the work, W, basedon first law of thermodynamics s :W=(H2 _H1)_Q (3.4.4)

Where H is the fluid enthalpy.When the initial conditions (pressure,temperature and entropy) and final pressuretogether with the polytropic constant are known, by combining equations 3.4.2 and3.4.3, the entropy at the final state s :

[1 Cl .Qxln(T2 Tl)J2=1 -1- (T2 T1)

(3.4.5)

The fluid temperaturecanthen be determined by performing a flash calculation at thefinal pressureand entropy.The following is the mathematical algorithm adoptedby Haque et al. :1) Selecta pressuredecrement.2) Expand the fluid in each zone polytropically by assuming C, and calculate the

energy Q which must be transferredto the fluid.

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4) Calculate the duration of the time-step correspondingto the chosen pressurestepby determining the flow rate through the choke.

5) Calculate all relevantheat transfer coefficients.6) Perform energy balanceson the fluid and vesselwall and calculate the energy Q*

transferredto the fluid.7) If Q# Q*, alter the value of C and return to step two. Otherwise proceed to step. eight.8) Perform mass balances on the fluid and calculate the quantity of liquid which

condensesand settles out from the gas and the quantity of gas which evaporatesout of the liquid hydrocarbon.

9) If depressurisation s complete, stop.Otherwisereturn to step one.HEAT AND MASS TRANSFER EFFECTS

Heat Transfer Between VesselWall/Fluid/SurroundingHeat transfer modes between zones 1 and 2 and the associated vessel wall weredescribed in section 3.4.1.1. When a significant quantity of water is encounteredduring blowdown, the authors concluded that based on experimental studies (resultsnot presented), the temperature of water varies relatively little and natural convectionis dominant. The correlations used to determine heat transfer coefficients forindividual phases are summarised in table 3.2. In zone 2, the maximum temperaturedifference between fluid and vessel wall for nucleate boiling and minimum one forfilm boiling are determined by standard correlations [Lienhard & Dhir, 1973;Berenson, 1961].

The overall energybalanceover the contentsand wall of the vessel aredeterminedbyfinite-difference solution of the transient heat conduction equation [Incropera, & DeWitt, 1985]. This method allows description of the temperature gradient across thewall thickness. As such, it can be applied to vessel wall with 'duplex construction.Heat transfer between the vesselwall and the surrounding is considered to be eitherby natural or forced convection depending on the nature of the surrounding and thecorresponding standard correlations [Perry & Chilton, 1973a, ; Churchill & Chu,1975] are used.ZMulti-layer construction

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corresponding standard correlations [Perry & Chilton, 1973a,b; Churchill & Chu,1975] are used.Table 3.2 Summary of heat transfer modes simulated by BLOWDOWN

[Haque et al., 1992a] for each fluid zone and the associated wallregion during blowdown

Zone Heat Transfer Mode Correlations Used to Determine Heat TransferCoefficients or Heat fluxes

Zone 1 Natural and forcedconvection wherenaturalconvection dominates.

Heat transfer coefficients arepredicted by :Natural convection : Perry & Chilton [1973a]Forced convection : Perry & Chilton [1973b]

Zone 2 Nucleate and film boiling Correspondingheat fluxes arepredicted by :NucleateBoiling : Rohsenow [1952]Film Boiling : Jordan [1968]Transition Boiling : No correlation is used.The heattransfer coefficients aredeterminedby linearinterpolation.

Zone 3 Natural convection Heat transfer coefficient is predicted from Perry &Chilton [1973a]

Heat and Mass Flux betweenAdjacent Fluid ZonesThe inter-phaseheat and mass fluxes between gaseous zone 1) and liquid (zone 2)hydrocarbons are a consequenceof evaporation and condensation of lighter andheavier componentsrespectively. In order to accurately describe the fluxes betweenthe corresponding phasesand the position of liquid droplets within the gas phase,information on phase equilibrium, nucleation time and settling velocity of liquiddroplets are required. The authors proposed the following empirical correlations todetermine the nucleation time, r (s), and the settling velocity of liquid droplets, v(ms-1) relative to the gas, as a function of the equilibrium liquid mole fraction, x inthe gasphase(zone 1):

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1T=- xv=0.03+3x

(3.4.6)

(3.4.7)

The above equationsare deducedfor mixtures of nitrogen, carbon dioxide, methane,ethane, propane and butane at pressuresup to 148 atm [Richardson, 1998]. Theauthors assumednegligible mass transfer between the liquid hydrocarbon (zone 2)and water (zone 3). However, heat transfer between the phasesoccurs because hephasesare not in general,at the samebulk temperature.The authors concluded that when the free water is warmer than the liquidhydrocarbon, heat transfer is mainly by boiling, otherwise it is by natural convection.Standardcorrelations are usedto determinethe inter-phaseheat transfer coefficients [see or example Perry & Chilton, 1973a; Rohsenow, 1952; Jordan, 1968; Lienhard &Dhir, 1973; Berenson, 1961].DISCHARGE RATE CALCULATION

A number of possible situations of the fluid statesat the entranceof and within thechoke were considered.The fluid approaching the choke can be one phase (vapour)or two-phase (liquid and vapour) and the fluid in the choke can be in a metastablestate or in thermodynamic and phase equilibrium. On the basesof their comparisonwith the experimental measurements, the best predictions were generated byassuming fluid phase equilibrium within the choke and at the entranceof the chokewhere the fluid could either be in one or two-phases.The rate of discharge through the choke is determined by assuming that the fluidaccelerates hrough the choke isentropically and the fluid velocity is equal to thelocal speed of sound when the back-pressure s sufficiently low. An energy balanceacross he choke is performed by assumingthe fluid velocity approaching the chokeis zero:

Hi -HC+ 12a2 (3.4.8)

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The local speedof sound,a, is given by :

FaP 2 Cp apa=Laps CVaPT(3.4.9)

All terms in the above equationare evaluatedunderchokedconditions. The abovetwo equationsare governedby the isentropic condition :

Si =Sc (3.4.10)Where Cp, CV = Specific heats at constant pressureand at constant volume

respectivelyHi, HC = Enthalpies of the fluid far upstreamof the orifice and at the

choke respectivelyP= PressureSi, SC = Entropies of the fluid far upstream of the orifice and at the

choke respectivelyS= Entropy of the fluidT= Temperaturep= Density

The conditions at the choke are calculated by solving equations 3.4.8 - 3.4.10iteratively. The mass flow rate is then determined from a knowledge of orifice areaand a dischargecoefficient.THERMOPHYSICAL PROPERTIES

The thermodynamic phase and transport properties are predicted by a computerpackage based on an extended principle of corresponding states [Rowlinson &Watson, 1969]. Briefly, this approach is based on relating the properties of themixture to those of methane chosenasa referencesubstance Saville & Szczepanski,1982]. The authors justified the choice of the extended principle of correspondingstates over the well known cubic equationsof states(CEOS) as a consequenceof itsaccurate epresentationof phase equilibrium, enthalpy and density, whilst the CEOS

1_ ,1

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are claimed to be much lesssuccessful n predicting the last two properties.However,the authors point out that solving the extended principle of corresponding states smore computationally demanding (ca. two folds [Richardson, 1997] ) compared tosolving a CEOS.EXPERIMENTAL VALIDATION

Predictions from BLOWDOWN were compared with experimental data. Threecomparisons were published, one with a small vessel containing nitrogen (initialpressure is 148 atm) and the other two with a full-size vessel containing differenthydrocarbon mixtures (initial pressures ca. 118.4 atm). In all cases, the modelaccurately predicted the pressure/time profiles as well as the minimum average bulkfluid and minimum inside wall temperatures to 5 K. The comparisons of fluid andinner wall temperature/time profiles for nitrogen were given in figure 2.1 in chapter2. The time profiles for the fluid and wall temperatures for a hydrocarbon mixture(66.5% mole methane, 3.5% mole ethane, 30.0% mole propane and traces of butanes)were respectively given in figures 2.2 and 2.3 in the same chapter.The corresponding results (with the exception of the measured liquid temperatures)for another hydrocarbon mixture containing 85.5% mole methane, 4.5% mole ethaneand 10.0% mole propane with traces of butanes are given in figures 3.12 and 3.13.Solid lines represent predictions from BLOWDOWN while the shaded regions showexperimental data.

From the above figures, although there is no comparison for liquid temperature,theproposed model is capable of predicting the variation of inner wetted walltemperature with time accurately (see figure 3.13). The authors indicated that thefluid in the vesselwas initially gas and liquid condensatestartedto form after about100 s, which was sometime after entering the mixture's phase envelope. Themaximum depth of liquid condensatewas about 0.1 m which was very low whencompared with the vesselheight of 3.24 m. They attributed the delay to the relativelyslow processof phaseequilibration. However, it is interesting to note that figure 3.13showing the measured and predicted temperatures of wall in contact with liquidphase ndicates the presenceof vapour condensatewell before 100 s.

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Additionally, the predicted liquid temperature (figure 3.12) initially decreasesandthen increasesat about 100 s before it decreases gain.This behaviour is very similarto the experimental observation made earlier by the authors [Haque et al., 1990] inthe caseof depressurisationof a nitrogen mixture (see igure 2.10 in chapter2) wherea small amount of vapour was condensedafter 10 s (total blowdown time was 100s).

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310

300

290

.280Co.m 270aE

260

I

Wall In contact with pas

250

0240

I

Wall in contact with liquid

I

200 400 600 800 1000 1200Time /s

Figure 3.13 Variations of inside wall temperatures with time fordepressurisation of a hydrocarbon mixture (85.5% mole methane,4.5 mole % ethane, 10.0 mole % propane with traces of highermolecular weight hydrocarbons in particular of butanes).Hatched regions span experimental measurements, solid lines arepredictions from BLOWDOWN [Haque et al. 1992b]

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3.4.2 Split Fluid Model [Overa et al., 19941In response o the shortcomings associatedwith the conventional methodsbasedoncommon engineering practice for blowdown (see section 3.2), Overa et al. [1994]proposed a SPLIT FLUID MODEL for blowdown of single or two-phasehydrocarbon mixtures. Figure 3.14 (page 71) is an overview of the depressurisationmodel and the various output,parameters.Calculations are carried out using variabletime steps hroughout the depressurisationprocess.The vessel s assumed o be at spatially uniform pressureand eachfluid zone is well-mixed. Hence, there is a uniform distribution of temperatureand composition in eachzone.Gasonly is assumed o be discharged rom the vessel.MATHEMATICAL ALGORITHM

Overa et al. [1994] divided the fluid within the vessel during blowdown into twophases:the liquid phasewhich comprises the existing liquid and condensedvapourfrom the sub-cooled vapour and the vapour phase which comprises the existingvapour and evaporated liquid from the boiling liquid. Heat transfer between thevapour and the liquid as well as the associated vessel walls are considered. Theauthors proposeddifferent thermodynamic trajectories for eachfluid phase.The enthalpy changeof the liquid phaseduring a given time interval is assumed o bedue to heat transfer effects only. Hence, work done by the liquid due to expansion signored. Heat transfer to the liquid phase s assumed o take place from wetted wallsurface. On the other hand, heat transfer from the vapour phase occurs due to thetemperaturedifference betweenthe two phases.The liquid temperature, TL, and the number of moles of evaporated liquid, Nb, atstage i+l are determined by pressureenthalpy flash at vessel pressure.The specificenthalpy of the liquid is given by :

Hii = NL HL (qL -gLV)Ot + Nc HNL +NC NL NL +NC LC(3.4.11)

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Where HLC, HL = Specific enthalpies of condensed vapour and liquidrespectively

NC, NL = Number of moles of condensed vapour and evaporatedliquid respectively

qL, qLV = Ratesof heattransferbetween iquid and vessel or vapourAt = Finite time stepfor calculation

The evaporated iquid is addedto the existing vapour and the vapour composition isupdated.The entropy change of vapour phase is determined on the basis of second law ofthermodynamics by assuming that the heat transfer during a given `small' timeinterval is infinitesimal and that there is no change n temperature.The vapour temperature, TV, vessel pressure, P, and the number of moles ofcondensedvapour, NC, at stage +1 are determined by a volume/entropy flash. Thevapour volume is fixed when both the vessel and liquid volumes are known. Thespecific entropy of the vapour is given by :

SV 1= NvNv +Nb

i (3.4.12)q y+Q LV )AtTN1+_Nb Ny +Nb Sb

Where Sb, Sr = Specific entropies of vaporized liquid and vapourrespectively

Nb, NV = Number of moles of vaporized liquid and vapourrespectively

qv = Rateof heat ransferbetweenvesselandvapourEquations3.4.11 and 3.4.12 may be solved iteratively until the vesselpressureduringa given time interval is found.

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Chapter 3 Review ofMathematical Modelsfor Blowdown Simulation

THERMOPHYSICAL PROPERTIES

The thermophysical properties are predicted by an in house process simulator[Wilson et al., 1991]. No correlation is specified.EXPERIMENTAL VALIDATION

Figures 2.6 in chapter 2 shows the variations of liquid and gas temperatureswithtime following the blowdown of a two-phase hydrocarbon mixture (no compositionspecified) from 19.7 atm and 25 C. Solid lines show datapredicted from the modelwhich are in good agreement with those obtained from experiment. The SPLITFLUID MODEL is capableof predicting temperatures o 4 K. The dashed ine showsthe resultsobtainedby assuming50% isentropic expansionof the fluid which greatlyoverestimates he fluid temperature.No data indicating the performanceof the modelin terms of predicting the pressure/timeprofile or the wall temperatures n contactwith the liquid and vapour phasesaregiven.

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Chapter.? Review of Mathematical Models for Blowdown Simulation

!1uTN

T,.N.

TLWToutside

Qambicnt

n1 FTv

MVC

7

Kees .MF.MVCMVDP

Mass of evaporated liquidMass of condensed vapourTotal mass of discharged vapourVessel pressure

P T,.

T,

,t,dr"

TL Liquid phase temperature

Qamtncnt Heat input (or loss) to ambientQL

QLVQV

Heat flow between vessel and liquidHeat flow between vapour and liquidHeat flow between vessel and vapour

QLNI

Wetted wall temperatureTemperature at the surface of the vessel

T, Vapour phase temperatureTv Discharge gas temperatureTv, Unwetted wall temperatureED0CD0

Vapour phaseLiquid phaseMetal wall layerInsulation wall layer

Figure 3.14 Depressurisation model for Split Fluid Model [Overa et. al., 19941

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3.5 CONCLUSION

On the basis of the literature reviewed in this chapter, it may be concluded thatsimple blowdown simulations basedon various engineering practices are hopelesslyinadequate n terms of predicting the minimum temperaturesencountered n practice.This is primarily becauseof the arbitrary choice of the thermodynamic trajectory ofthe fluid during blowdown. The appropriate thermodynamic trajectory shouldquantitatively account for heat transfer effects taking place between the fluid phasesandassociatedwalls.So far, the BLOWDOWN model, developed by Haque et at. [1992a] is the mostsophisticated model available for blowdown simulation as it models all thesignificant effects taking place during blowdown. However, the use of an extendedprinciple of corresponding statesfor predicting thermophysical properties may giverise to significant computational work load and consistency problems in practicalsituations as it is rarely employed in the process environment. Although it is claimedthat the commonly used cubic equationsof statewill introduce error to predictions ofthe required thermophysical data, especially liquid density, their effect on theperformanceof the blowdown simulation is not known.Despite the simplicity of Overa et al. s [1994] blowdown model, it has not beenvalidated at high pressures (>19.7 atm) for blowdown of hydrocarbon mixtures.Additionally, the authors use an experimentally determined heat transfer coefficientbetween the wetted wall and the liquid phase and hence the model is not strictlypredictive. Furthermore, the model is not capable of predicting the presenceof two-phasedischarge at the orifice; it assumesvapour discharge only. The effects of thisassumption on the predictions of temperature and pressure during blowdownespecially at elevated pressuresarenot known.In conclusion, it is obvious, based on mathematical models by Overa et al. [1994]and Haque et al. [1992a], that the thermodynamic trajectory for vapour phaseneedsto account for both work and heat. However, the authors have defined differentrelationships between heat and temperature. Overa et al. assume the heat transfer

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Chapter 4 The Developmentof a Blowdown Mathematical Model : BLOWSIM

CHAPTER 4THE DEVELOPMENT OF A BLOWDOWN MATHEMATICALMODEL : BLOWSIM4.1 INTRODUCTION

The review of the important mathematical models for blowdown, particularlyBLOWDOWN [Haque et al., 1992a] and SPLIT FLUID MODEL [Overa et al.,1994], highlighted the following points which require further investigation:

1) The effects of cubic equations of state on the performance of blowdownsimulation in terms of predicting field data.

2) The suitability of Overa et al. s thermodynamic trajectory for vapour phaseexpansionduring blowdown of hydrocarbonmixtures from elevatedpressures.

3) The effects of applying different thermodynamic trajectories to liquid phase nterms of the accuracy in predicting temperatures and pressures duringblowdown.

4) The effect of Overa et al. s [1994] suggestedconstantheat transfer coefficientsbetween liquid and wetted wall compared to the boiling heat flux empiricalcorrelation employed by Haque et al. [1992a] on the accuracy of the predictedliquid and wetted wall temperatures.

5) Comparisons of the rigorous [Haque et al., 1992a] against simple dischargecalculation methods (based on ideal gas assumption) in terms of predictingtemperaturesandpressuresduring blowdown.

6) The evelof sophisticationequiredn blowdownmodelling.This chapterdescribes he development of a mathematical model, BLOWSIM whichaddresses he above issues. It allows for vapour spaceblowdown through a singleorifice from the top of an isolated vessel containing single (vapour) or two-phase(vapour and liquid) hydrocarbon mixtures. The evaluation of performance of variouscubic equations of state in predicting the appropriate thermodynamic properties aredescribed in the next chapter. An evaluation of the accuracy of the model by

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Chapter 4 The Developmentof a Blowdown Mathematical Model: BLOWSIM

comparison with experimental data and those predicted from BLOWDOWN ispresented n chapter6.4.2 DEVELOPMENT OF BLOWSIM MATHEMATICAL MODEL

BLOWSIM accountsfor the most important processesdescribed n chapter 2 whichtake place during blowdown. These include non-equilibrium effects betweenphases,heat transfer between each fluid phase and their corresponding sections of vesselwall, inter-phasefluxes due to evaporationand condensation, and the effects of sonicflow at the orifice. Typical output include the variations of discharge rate, pressuretogetherwith fluid and wall temperatureswith time.The temperature and composition in each fluid phase are assumedto be spatiallyuniform (well mixed), and the pressure is assumed o be spatially uniform in thevessel. Although the results of published blowdown experiments [Haque et al.,1992b] indicate the presence of temperature gradients in both vapour and liquidphases the effect is more pronounced in vapour phase; figure 2.2, chapter 2), Haqueet al. [1992b] and Overa et al. [1994] demonstrated hat their blowdown simulationsbasedon the above assumptions gave reasonablepredictions when compared withexperimental data (see figures 2.1- 2.3 for Haque et al. s validations and figure 2.6for Overa et al. s validation). In addition, the fluid prior to blowdown is assumed obe at equilibrium andthe vesselwall temperatureequal to the fluid temperature.The detailed assumptions regarding heat and mass transfer together with dischargecalculations aredescribed ater in sections4.2.2 - 4.2.5.4.2.1 Application of Finite Difference Method to Blowdown CalculationThe equations to be solved in blowdown calculation are mainly non-linear andfunctions of both pressure and time. It is more practical to solve these equationsbyfinite difference method instead of integration. In this study, we approximate thedepressurisationprocessby a seriesof variable pressure ncrements. The advantagesof employing pressure increments are it s thermodynamic convenience andcomputational efficiency when compared to choosing time intervals (employed byOvera et al. s model [1994] ) which arethermodynamically irrelevant.

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Chapter 4 The Development of a Bl

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